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Theorem ldualgrplem 28708
Description: Lemma for ldualgrp 28709. (Contributed by NM, 22-Oct-2014.)
Hypotheses
Ref Expression
ldualgrp.d  |-  D  =  (LDual `  W )
ldualgrp.w  |-  ( ph  ->  W  e.  LMod )
ldualgrp.v  |-  V  =  ( Base `  W
)
ldualgrp.p  |-  .+  =  o F ( +g  `  W
)
ldualgrp.f  |-  F  =  (LFnl `  W )
ldualgrp.r  |-  R  =  (Scalar `  W )
ldualgrp.k  |-  K  =  ( Base `  R
)
ldualgrp.t  |-  .X.  =  ( .r `  R )
ldualgrp.o  |-  O  =  (oppr
`  R )
ldualgrp.s  |-  .x.  =  ( .s `  D )
Assertion
Ref Expression
ldualgrplem  |-  ( ph  ->  D  e.  Grp )

Proof of Theorem ldualgrplem
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ldualgrp.f . . . 4  |-  F  =  (LFnl `  W )
2 ldualgrp.d . . . 4  |-  D  =  (LDual `  W )
3 eqid 2283 . . . 4  |-  ( Base `  D )  =  (
Base `  D )
4 ldualgrp.w . . . 4  |-  ( ph  ->  W  e.  LMod )
51, 2, 3, 4ldualvbase 28689 . . 3  |-  ( ph  ->  ( Base `  D
)  =  F )
65eqcomd 2288 . 2  |-  ( ph  ->  F  =  ( Base `  D ) )
7 eqidd 2284 . 2  |-  ( ph  ->  ( +g  `  D
)  =  ( +g  `  D ) )
8 eqid 2283 . . 3  |-  ( +g  `  D )  =  ( +g  `  D )
943ad2ant1 976 . . 3  |-  ( (
ph  /\  x  e.  F  /\  y  e.  F
)  ->  W  e.  LMod )
10 simp2 956 . . 3  |-  ( (
ph  /\  x  e.  F  /\  y  e.  F
)  ->  x  e.  F )
11 simp3 957 . . 3  |-  ( (
ph  /\  x  e.  F  /\  y  e.  F
)  ->  y  e.  F )
121, 2, 8, 9, 10, 11ldualvaddcl 28693 . 2  |-  ( (
ph  /\  x  e.  F  /\  y  e.  F
)  ->  ( x
( +g  `  D ) y )  e.  F
)
13 ldualgrp.r . . . . 5  |-  R  =  (Scalar `  W )
14 eqid 2283 . . . . 5  |-  ( +g  `  R )  =  ( +g  `  R )
154adantr 451 . . . . 5  |-  ( (
ph  /\  ( x  e.  F  /\  y  e.  F  /\  z  e.  F ) )  ->  W  e.  LMod )
16 simpr2 962 . . . . 5  |-  ( (
ph  /\  ( x  e.  F  /\  y  e.  F  /\  z  e.  F ) )  -> 
y  e.  F )
17 simpr3 963 . . . . 5  |-  ( (
ph  /\  ( x  e.  F  /\  y  e.  F  /\  z  e.  F ) )  -> 
z  e.  F )
181, 13, 14, 2, 8, 15, 16, 17ldualvadd 28692 . . . 4  |-  ( (
ph  /\  ( x  e.  F  /\  y  e.  F  /\  z  e.  F ) )  -> 
( y ( +g  `  D ) z )  =  ( y  o F ( +g  `  R
) z ) )
1918oveq2d 5874 . . 3  |-  ( (
ph  /\  ( x  e.  F  /\  y  e.  F  /\  z  e.  F ) )  -> 
( x  o F ( +g  `  R
) ( y ( +g  `  D ) z ) )  =  ( x  o F ( +g  `  R
) ( y  o F ( +g  `  R
) z ) ) )
20 simpr1 961 . . . 4  |-  ( (
ph  /\  ( x  e.  F  /\  y  e.  F  /\  z  e.  F ) )  ->  x  e.  F )
211, 2, 8, 15, 16, 17ldualvaddcl 28693 . . . 4  |-  ( (
ph  /\  ( x  e.  F  /\  y  e.  F  /\  z  e.  F ) )  -> 
( y ( +g  `  D ) z )  e.  F )
221, 13, 14, 2, 8, 15, 20, 21ldualvadd 28692 . . 3  |-  ( (
ph  /\  ( x  e.  F  /\  y  e.  F  /\  z  e.  F ) )  -> 
( x ( +g  `  D ) ( y ( +g  `  D
) z ) )  =  ( x  o F ( +g  `  R
) ( y ( +g  `  D ) z ) ) )
231, 2, 8, 15, 20, 16ldualvaddcl 28693 . . . . 5  |-  ( (
ph  /\  ( x  e.  F  /\  y  e.  F  /\  z  e.  F ) )  -> 
( x ( +g  `  D ) y )  e.  F )
241, 13, 14, 2, 8, 15, 23, 17ldualvadd 28692 . . . 4  |-  ( (
ph  /\  ( x  e.  F  /\  y  e.  F  /\  z  e.  F ) )  -> 
( ( x ( +g  `  D ) y ) ( +g  `  D ) z )  =  ( ( x ( +g  `  D
) y )  o F ( +g  `  R
) z ) )
251, 13, 14, 2, 8, 15, 20, 16ldualvadd 28692 . . . . 5  |-  ( (
ph  /\  ( x  e.  F  /\  y  e.  F  /\  z  e.  F ) )  -> 
( x ( +g  `  D ) y )  =  ( x  o F ( +g  `  R
) y ) )
2625oveq1d 5873 . . . 4  |-  ( (
ph  /\  ( x  e.  F  /\  y  e.  F  /\  z  e.  F ) )  -> 
( ( x ( +g  `  D ) y )  o F ( +g  `  R
) z )  =  ( ( x  o F ( +g  `  R
) y )  o F ( +g  `  R
) z ) )
2713, 14, 1, 15, 20, 16, 17lfladdass 28636 . . . 4  |-  ( (
ph  /\  ( x  e.  F  /\  y  e.  F  /\  z  e.  F ) )  -> 
( ( x  o F ( +g  `  R
) y )  o F ( +g  `  R
) z )  =  ( x  o F ( +g  `  R
) ( y  o F ( +g  `  R
) z ) ) )
2824, 26, 273eqtrd 2319 . . 3  |-  ( (
ph  /\  ( x  e.  F  /\  y  e.  F  /\  z  e.  F ) )  -> 
( ( x ( +g  `  D ) y ) ( +g  `  D ) z )  =  ( x  o F ( +g  `  R
) ( y  o F ( +g  `  R
) z ) ) )
2919, 22, 283eqtr4rd 2326 . 2  |-  ( (
ph  /\  ( x  e.  F  /\  y  e.  F  /\  z  e.  F ) )  -> 
( ( x ( +g  `  D ) y ) ( +g  `  D ) z )  =  ( x ( +g  `  D ) ( y ( +g  `  D ) z ) ) )
30 eqid 2283 . . . 4  |-  ( 0g
`  R )  =  ( 0g `  R
)
31 ldualgrp.v . . . 4  |-  V  =  ( Base `  W
)
3213, 30, 31, 1lfl0f 28632 . . 3  |-  ( W  e.  LMod  ->  ( V  X.  { ( 0g
`  R ) } )  e.  F )
334, 32syl 15 . 2  |-  ( ph  ->  ( V  X.  {
( 0g `  R
) } )  e.  F )
344adantr 451 . . . 4  |-  ( (
ph  /\  x  e.  F )  ->  W  e.  LMod )
3533adantr 451 . . . 4  |-  ( (
ph  /\  x  e.  F )  ->  ( V  X.  { ( 0g
`  R ) } )  e.  F )
36 simpr 447 . . . 4  |-  ( (
ph  /\  x  e.  F )  ->  x  e.  F )
371, 13, 14, 2, 8, 34, 35, 36ldualvadd 28692 . . 3  |-  ( (
ph  /\  x  e.  F )  ->  (
( V  X.  {
( 0g `  R
) } ) ( +g  `  D ) x )  =  ( ( V  X.  {
( 0g `  R
) } )  o F ( +g  `  R
) x ) )
3831, 13, 14, 30, 1, 34, 36lfladd0l 28637 . . 3  |-  ( (
ph  /\  x  e.  F )  ->  (
( V  X.  {
( 0g `  R
) } )  o F ( +g  `  R
) x )  =  x )
3937, 38eqtrd 2315 . 2  |-  ( (
ph  /\  x  e.  F )  ->  (
( V  X.  {
( 0g `  R
) } ) ( +g  `  D ) x )  =  x )
40 eqid 2283 . . 3  |-  ( inv g `  R )  =  ( inv g `  R )
41 eqid 2283 . . 3  |-  ( z  e.  V  |->  ( ( inv g `  R
) `  ( x `  z ) ) )  =  ( z  e.  V  |->  ( ( inv g `  R ) `
 ( x `  z ) ) )
4231, 13, 40, 41, 1, 34, 36lflnegcl 28638 . 2  |-  ( (
ph  /\  x  e.  F )  ->  (
z  e.  V  |->  ( ( inv g `  R ) `  (
x `  z )
) )  e.  F
)
431, 13, 14, 2, 8, 34, 42, 36ldualvadd 28692 . . 3  |-  ( (
ph  /\  x  e.  F )  ->  (
( z  e.  V  |->  ( ( inv g `  R ) `  (
x `  z )
) ) ( +g  `  D ) x )  =  ( ( z  e.  V  |->  ( ( inv g `  R
) `  ( x `  z ) ) )  o F ( +g  `  R ) x ) )
4431, 13, 40, 41, 1, 34, 36, 14, 30lflnegl 28639 . . 3  |-  ( (
ph  /\  x  e.  F )  ->  (
( z  e.  V  |->  ( ( inv g `  R ) `  (
x `  z )
) )  o F ( +g  `  R
) x )  =  ( V  X.  {
( 0g `  R
) } ) )
4543, 44eqtrd 2315 . 2  |-  ( (
ph  /\  x  e.  F )  ->  (
( z  e.  V  |->  ( ( inv g `  R ) `  (
x `  z )
) ) ( +g  `  D ) x )  =  ( V  X.  { ( 0g `  R ) } ) )
466, 7, 12, 29, 33, 39, 42, 45isgrpd 14507 1  |-  ( ph  ->  D  e.  Grp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {csn 3640    e. cmpt 4077    X. cxp 4687   ` cfv 5255  (class class class)co 5858    o Fcof 6076   Basecbs 13148   +g cplusg 13208   .rcmulr 13209  Scalarcsca 13211   .scvsca 13212   0gc0g 13400   Grpcgrp 14362   inv gcminusg 14363  opprcoppr 15404   LModclmod 15627  LFnlclfn 28620  LDualcld 28686
This theorem is referenced by:  ldualgrp  28709
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-plusg 13221  df-sca 13224  df-vsca 13225  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-lmod 15629  df-lfl 28621  df-ldual 28687
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